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Theorem fliftf 5668
Description: The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
flift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
flift.3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
Assertion
Ref Expression
fliftf  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
Distinct variable groups:    x, R    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftf
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  Fun  F )
2 flift.1 . . . . . . . . . . 11  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
3 flift.2 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
4 flift.3 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
52, 3, 4fliftel 5662 . . . . . . . . . 10  |-  ( ph  ->  ( y F z  <->  E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
65exbidv 1781 . . . . . . . . 9  |-  ( ph  ->  ( E. z  y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
76adantr 274 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
8 rexcom4 2683 . . . . . . . . 9  |-  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) )
9 elisset 2674 . . . . . . . . . . . . . 14  |-  ( B  e.  S  ->  E. z 
z  =  B )
104, 9syl 14 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  E. z 
z  =  B )
1110biantrud 302 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
y  =  A  <->  ( y  =  A  /\  E. z 
z  =  B ) ) )
12 19.42v 1862 . . . . . . . . . . . 12  |-  ( E. z ( y  =  A  /\  z  =  B )  <->  ( y  =  A  /\  E. z 
z  =  B ) )
1311, 12syl6rbbr 198 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  ( E. z ( y  =  A  /\  z  =  B )  <->  y  =  A ) )
1413rexbidva 2411 . . . . . . . . . 10  |-  ( ph  ->  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A ) )
1514adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  Fun  F )  ->  ( E. x  e.  X  E. z
( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
168, 15syl5bbr 193 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z E. x  e.  X  ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
177, 16bitrd 187 . . . . . . 7  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. x  e.  X  y  =  A ) )
1817abbidv 2235 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  { y  |  E. z  y F z }  =  {
y  |  E. x  e.  X  y  =  A } )
19 df-dm 4519 . . . . . 6  |-  dom  F  =  { y  |  E. z  y F z }
20 eqid 2117 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
2120rnmpt 4757 . . . . . 6  |-  ran  (
x  e.  X  |->  A )  =  { y  |  E. x  e.  X  y  =  A }
2218, 19, 213eqtr4g 2175 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  dom  F  =  ran  ( x  e.  X  |->  A ) )
23 df-fn 5096 . . . . 5  |-  ( F  Fn  ran  ( x  e.  X  |->  A )  <-> 
( Fun  F  /\  dom  F  =  ran  (
x  e.  X  |->  A ) ) )
241, 22, 23sylanbrc 413 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  F  Fn  ran  ( x  e.  X  |->  A ) )
252, 3, 4fliftrel 5661 . . . . . . 7  |-  ( ph  ->  F  C_  ( R  X.  S ) )
2625adantr 274 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  F  C_  ( R  X.  S ) )
27 rnss 4739 . . . . . 6  |-  ( F 
C_  ( R  X.  S )  ->  ran  F 
C_  ran  ( R  X.  S ) )
2826, 27syl 14 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  ran  ( R  X.  S
) )
29 rnxpss 4940 . . . . 5  |-  ran  ( R  X.  S )  C_  S
3028, 29sstrdi 3079 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  S
)
31 df-f 5097 . . . 4  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  <->  ( F  Fn  ran  ( x  e.  X  |->  A )  /\  ran  F 
C_  S ) )
3224, 30, 31sylanbrc 413 . . 3  |-  ( (
ph  /\  Fun  F )  ->  F : ran  ( x  e.  X  |->  A ) --> S )
3332ex 114 . 2  |-  ( ph  ->  ( Fun  F  ->  F : ran  ( x  e.  X  |->  A ) --> S ) )
34 ffun 5245 . 2  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  ->  Fun  F )
3533, 34impbid1 141 1  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   E.wrex 2394    C_ wss 3041   <.cop 3500   class class class wbr 3899    |-> cmpt 3959    X. cxp 4507   dom cdm 4509   ran crn 4510   Fun wfun 5087    Fn wfn 5088   -->wf 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  qliftf  6482
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